London to Milan - Discount Airline?

Don't be nervous. Budget airlines are what the British call "cheap and cheerful". I.e. they will get you from A to B for the price you pay, but don't expect anything else. If they don't, consumer protection authorities (national and EU) are down on them "like a ton of bricks". Millions use them every year. Just make sure you turn up on time, and ensure you read and abide by their check-in and luggage rules.
Except, beware of Ryanair. they have a reputation of being cheap-and-nasty.
Except, check which airport you are flying into and cost in the transfer. Especially Ryanair, which has a reputation for flying to "any old strip of concrete in the middle of nowhere, so long as it is cheap".
The "tradional airlines" can also offer cheap fares if you book in advance, because they have to compete with the budget airlines.

Navigating a country when you don't know the language is just a skill you have to learn. Millions of Europenas have already learnt it. Just don't expect everybody to speak fluent Englush - that is plain rude. But, luckily for you, English is the language taught in schools in most countries. So a Swede talking to an Italian will speak to each other in school basic English. Nobody objects because it is a "foreign language" to both of them, and both make mistakes which the other ignores. Just assume the other person speaks the same level of English as you do whatever-language-you-learnt-in-school, bring a phrase book (for menus etc.), and be hyper polite, and you will be OK.

look on www.skyscanner.com for cheap flights. Malpensa or Linate airports are both good options

We've flown easyJet from Rome to Palermo, Italy, Vueling from Bilbao, Spain to Barcelona, and others between Romania & Italy and Spain & Portugal. Sometimes the budget planes are smaller than with bigger airlines, have few (if any) frills on board, and can have strict baggage policies (as mentioned above, read their fine print about baggage), but they get you to your destination safely and for a good price. Good luck with your studies and have a great visit in Italy!

Whenever I want to know what airlines fly non-stop out of any airport, I go to Wikipedia and put in the airport's name.

Gatwick Airport has budget air carriers flying all over the continent. I'm always thrilled to get a $100 one way flight anywhere. Many of the budget carriers' airfares are surprisingly cheap--but go up in times of high demand.

We flew EasyJet.com from Gatwick to Pisa--on the way to Florence.

Milan is a very busy international airport--#2 in Italy for flights.

You've gotten a lot of good advice above.

If you are interested in going by train, here's how, from The Man In Seat 61: http://www.seat61.com/Italy.htm#London%20to%20Bologna,%20Florence,%20Rome,%20Naples

For a trip of that distance, I'd fly. Skyscanner is the way to find all your options. Be sure to select "all London airports" and "all Milan airports" to see all the options (London has 6 airports and Milan has 3, at least by the way the airlines define it). For London, Heathrow is closest; for Milan, Linate is closest. If you're using one of the others, see how much time and money is involved in getting from the city to the airport, and make sure the savings are worth the hassle. For Milan, Malpensa and Bergamo (the not-close ones) have regular bus service to the city, taking about an hour; Malpensa also has a train taking about 40 minutes. So neither of these is a real problem. I don't know the latest details about getting to and from London's airports.

If you want to fly to Rome instead, there are two airports (Fiumicino, the "main" one, and Ciampino). Either works; both have buses to the center, and Fiumicino also has a direct train.

And yes, while Ryanair has a "reputation," the other budget carriers are quite reasonable. But with any of them, be SURE you read and understand all the rules; they WILL be enforced, as that's how they make their money. For instance, if they say that any bags over 10 kilos must be checked, don't think you can get away with 10.5 kilos as carry-on; they will make you check it and charge you the higher fee for not having paid for a checked bag in advance. And as said above, do look at the "legacy" carriers like British Air and Alitalia; they may not cost more, particularly if they will check bags for free (or let you carry them on) that you would have to pay to check on a budget carrier. They will also save money if they use an airport that costs less to get to (for instance, you can get to Heathrow on the tube, but the other London airports require a train or bus).

Here's a fun musical guide to the potential perils of Cheap Flights (not safe for work-contains profanity): http://tinyurl.com/2vxmaod

As for navigating Italy, many have done it without speaking Italian. Do your research beforehand; look at guidebooks and websites so you know what to expect. Be sure to look at Rick Steves Italy, as he has lots of great tips that will make a first visit easier. And if you can learn a few basics of Italian, it will make your trip easier. You can get a single CD like Berlitz Italian (comes with handy phrasebook) as a great starter.